Difference between revisions of "2023 AMC 12B Problems/Problem 12"

Revision as of 16:52, 15 November 2023

For complex number $u = a+bi$ and $v = c+di$ (where $i=\sqrt{-1}$ ), define the binary operation

$u \odot v = ac + bdi$

Suppose $z$ is a complex number such that $z\odot z = z^{2}+40$ . What is $|z|$ ?

$\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }\sqrt{5}\qquad\textbf{(D) }\sqrt{10}\qquad\textbf{(E) }5\sqrt{2}$

@@ Line 3: / Line 3: @@
 For complex number <math>u = a+bi</math> and <math>v = c+di</math> (where <math>i=\sqrt{-1}</math>), define the binary operation
-<math>u \xdot v = ac + bdi</math>
+<math>u \odot v = ac + bdi</math>
-Suppose <math>z</math> is a complex number such that <math>z\xdot z = z^{2}+40</math>. What is <math>|z|</math>?
+Suppose <math>z</math> is a complex number such that <math>z\odot z = z^{2}+40</math>. What is <math>|z|</math>?
 <math>\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }\sqrt{5}\qquad\textbf{(D) }\sqrt{10}\qquad\textbf{(E) }5\sqrt{2}</math>